Smallest Multiples for Perfect Cubes
Class 8 Mathematics | Topic 6.5
Learning Objective
Students will be able to find the smallest multiples of numbers that make perfect cubes through prime factorization.
Time Allotment
45 minutes
Materials Needed
Textbook, worksheets, cubes for visualization
Hook Activity (4 Minutes)
Begin with a visual demonstration using the 3D cube below. Ask students:
- What makes a cube different from other 3D shapes?
- How many smaller cubes would we need to build a larger cube?
Activate prior knowledge by asking students to recall what they know about perfect squares and how they relate to perfect cubes.
Explicit Teaching/Teacher Modelling (8 Minutes)
Demonstrate how to determine if a number is a perfect cube using prime factorization.
Example from Textbook:
Raj made a cuboid of plasticine with dimensions 15 cm × 30 cm × 15 cm.
Volume = 15 × 30 × 15 = 3 × 5 × 2 × 3 × 5 × 3 × 5 = 2 × 3³ × 5³
Since there's only one '2' in the prime factorization, we need 2 × 2 = 4 such cuboids to make a perfect cube.
Example 2: Is 392 a perfect cube?
392 = 2 × 2 × 2 × 7 × 7
The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect cube.
To make it a cube, we need one more 7: 392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 (which is 14³)
Group Work (We Do - 16 Minutes)
Divide students into groups of 4-5. Provide each group with different numbers between 1-100.
Tasks for each group:
- Identify which numbers are perfect cubes
- For non-perfect cubes, find the smallest number to multiply or divide to make them perfect cubes
- Present findings to the class
Try These Examples in Your Groups:
1. Is 2700 a perfect cube? If not, what is the smallest number to multiply to make it a perfect cube?
2700 = 2² × 3³ × 5²
We need one more 2 and one more 5 to make groups of three.
Smallest number to multiply: 2 × 5 = 10
2700 × 10 = 27000 = 30³
2. Is 16000 a perfect cube? If not, what is the smallest number to divide to make it a perfect cube?
16000 = 2⁷ × 5³
We have extra 2's (7 instead of a multiple of 3).
Smallest number to divide: 2¹ = 2 (since 7-6=1 extra 2)
16000 ÷ 2 = 8000 = 20³
Independent Work (You Do - 8 Minutes)
Students work individually on these problems:
1. Is 64000 a perfect cube? Verify your answer.
64000 = 2⁸ × 5³
8 is not a multiple of 3, so 64000 is not a perfect cube.
We need to multiply by 2¹ = 2 to make the exponent of 2 a multiple of 3 (9).
64000 × 2 = 128000 = (2³ × 5)³ = (40)³
2. Is 900 a perfect cube? If not, find the smallest natural number to multiply.
900 = 2² × 3² × 5²
None of the prime factors appear in groups of three.
We need one more 2, one more 3, and one more 5.
Smallest number to multiply: 2 × 3 × 5 = 30
900 × 30 = 27000 = 30³
Closure (4 Minutes)
Summarize the key concepts:
- A perfect cube has prime factors in groups of three
- To make a number a perfect cube, multiply or divide by the smallest number needed to complete these groups
- The process involves prime factorization and understanding exponents
Think, Discuss and Write
Check which of the following are perfect cubes: 2700, 16000, 64000, 900, 125000, 36000, 21600, 10000, 27000000, 1000.
What pattern do you observe in these perfect cubes?
Perfect cubes from the list: 64000, 125000, 27000000, 1000
Pattern: In the prime factorization of a perfect cube, every prime factor has an exponent that is a multiple of 3.
Assessment Questions
Factual Questions
1. How is a cube different from a square?
A square is a 2-dimensional shape with four equal sides and four right angles. A cube is a 3-dimensional shape with six equal square faces, twelve equal edges, and eight vertices.
2. Compute the cube of 64.
64³ = 64 × 64 × 64 = 262,144
3. What is the story of S. Ramanujan and Prof. G. H. Hardy about?
The famous story involves Hardy visiting Ramanujan in the hospital. Hardy mentioned that the taxi he arrived in had the number 1729, which he thought was a dull number. Ramanujan immediately responded that 1729 is actually very interesting as it is the smallest number that can be expressed as the sum of two cubes in two different ways: 1729 = 1³ + 12³ = 9³ + 10³.
4. How do you differentiate between prime factorization of numbers and prime factorization of cubes?
In the prime factorization of any number, the exponents of prime factors can be any natural number. However, in the prime factorization of a perfect cube, every prime factor must have an exponent that is a multiple of 3 (i.e., each prime factor appears in groups of three).
Open-ended Questions
1. If a number is a perfect cube, what can you say about its prime factors?
If a number is a perfect cube, then in its prime factorization, every prime factor appears with an exponent that is a multiple of 3. This means each prime factor is present in groups of three. For example, 216 = 2³ × 3³ = (2 × 3)³ = 6³, so 216 is a perfect cube.
2. Why is it important to understand patterns in mathematics?
Understanding patterns in mathematics is crucial because:
- Patterns help us make predictions and solve problems more efficiently
- They reveal the underlying structure of mathematical concepts
- Recognizing patterns allows us to generalize concepts and create formulas
- Patterns make mathematics more intuitive and less about memorization
- They help connect different mathematical concepts and see the bigger picture
3. Can you create your own example of adding consecutive odd numbers to find a cube?
Yes! The sum of consecutive odd numbers can form perfect cubes. For example:
1 = 1³
3 + 5 = 8 = 2³
7 + 9 + 11 = 27 = 3³
13 + 15 + 17 + 19 = 64 = 4³
My own example: 21 + 23 + 25 + 27 + 29 = 125 = 5³
This pattern shows that the sum of n consecutive odd numbers starting from n² - n + 1 equals n³.
